The-Wolf,
All formal taxa are capitalised with the sole exception of species names (and subspecies names, if used). Hence: Animalia, Chordata, Vertebrata, Mammalia, Felidae; Felis; catus = Felis catus, the house cat. Subspecies names are also lower case, hence Felis silvestris libyca is the African wild cat, by contrast with the European wild cat Felis silvestris silvestris.
The above is true for animal names and fossil names. Things may be slightly different for plants and bacteria, which have their own rules. But I cannot speak with authority on them!
Normally, informal taxa are not capitalised. These are the collective names based on taxa (such as: crustacean, based on Crustacea; or hominid, based on Hominidae).
Common names are typically -- but not universally -- not capitalised. I was certainly taught not to in my undergraduate degree in zoology. But some publishers of scientific and popular journals capitalise common names. To take one example close to our hearts, the British magazine PFK (inconsistently as far as I can tell) capitalises common names, i.e., Neon Tetra. By contrast, TFH does not, and explicitly tells authors not to do so. The argument made by advocates of "capitalised common names" is that it makes it clearer what species are being discussed. In a scientific paper, the italicisation for species makes them stand out in a body of text; in a popular article, common names are used, and capitalising them makes them stand out instead.
Cheers, Neale